= Solution
Set $u_\theta=0$. Continuity gives $\partial_r(ru_r)=0$, hence $\boxed{u_r=g(\theta)/r}$. Substituting into radial and angular momentum gives
$$
p_r=\frac\rho{r^3}(g^2+\nu g''),\qquad p_\theta=\frac{2\rho\nu}{r^2}g'.
$$
For example $\nabla^2u_r-u_r/r^2=g''/r^3$, and the radial acceleration is $-g^2/r^3$. Equality of the mixed <pressure> derivatives now requires
$$
\rho(2gg'+\nu g^{(3)})/r^3=-4\rho\nu g'/r^3.
$$
Thus the equation for <radial viscous flow in a wedge> is
$$
\boxed{\nu(g^{(3)}+4g')+2gg'=0.}
$$
It can also be integrated once as $\nu(g''+4g)+g^2=\text{constant}$.
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